A norma ABNT ISO/IEC GUIA 98-3 é um guia que aborda a expressão da incerteza de medição, fornecendo diretrizes para a estimativa e a apresentação dos resultados de medição de forma confiável e consistente. Seguir essa norma é de extrema importância para garantir a qualidade e a precisão das medições realizadas, bem como para a devida interpretação dos resultados obtidos.
A ABNT ISO/IEC GUIA 98-3 baseia-se no documento GUM (Guide to the Expression of Uncertainty in Measurement), de 1995, e estabelece uma metodologia padronizada para a avaliação e a expressão da incerteza de medição. A incerteza de medição é uma medida da variabilidade e da falta de conhecimento completo associadas a uma medição. Ao seguir as diretrizes do guia, os profissionais podem estimar e comunicar a incerteza de medição de forma precisa e coerente.
Uma das principais vantagens de seguir a ABNT ISO/IEC GUIA 98-3 é a obtenção de resultados confiáveis e comparáveis. Ao utilizar a metodologia estabelecida no guia, os laboratórios e os profissionais de medição podem identificar e considerar todas as fontes de incerteza relevantes, desde as condições ambientais até as características dos instrumentos utilizados. Isso resulta em resultados mais precisos e confiáveis, aumentando a credibilidade das medições realizadas.
Além disso, a norma contribui para a tomada de decisões informadas com base nos resultados de medição. A expressão correta da incerteza permite que os usuários dos resultados compreendam a faixa de variação associada às medições e tomem decisões considerando essa incerteza. Isso é especialmente relevante em áreas como a indústria, a saúde e o meio ambiente, onde decisões importantes são baseadas em resultados de medição. A consideração da incerteza auxilia na avaliação da conformidade com os requisitos e na tomada de medidas apropriadas.
Outro aspecto relevante é a melhoria da qualidade dos processos de medição. Ao seguir a ABNT ISO/IEC GUIA 98-3, os profissionais e os laboratórios são incentivados a realizar uma avaliação cuidadosa das fontes de incerteza e a implementar ações corretivas para reduzir a incerteza de medição sempre que possível. Isso promove a melhoria contínua dos processos de medição, aumentando a confiabilidade dos resultados e a eficiência dos laboratórios.
Por fim, a norma também contribui para a conformidade com requisitos regulatórios e normativos. Em muitos setores, existem requisitos específicos relacionados à expressão da incerteza de medição, especialmente em áreas como a metrologia, a indústria farmacêutica e os ensaios laboratoriais. Seguir a ABNT ISO/IEC GUIA 98-3 garante que os laboratórios estejam em conformidade com esses requisitos, evitando problemas legais e regulatórios.
Em resumo, seguir a norma ABNT ISO/IEC GUIA 98-3 é fundamental para a expressão adequada da incerteza de medição. Ela fornece diretrizes claras e consistentes para a estimativa e a apresentação da incerteza, resultando em medições mais confiáveis, decisões informadas, melhoria da qualidade dos processos de medição e conformidade com requisitos regulatórios. A aplicação correta da norma contribui para a qualidade e a credibilidade das medições realizadas em diversos campos de atuação.
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Uncertainty of measurement
Part 3:
Guide to the expression of uncertainty in measurement
(GUM:1995)
Supplement 1:
Propagation of distributions using a Monte Carlo method
1 Scope
This Supplement provides a general numerical approach, consistent with the broad principles of the GUM [ISO/IEC Guide 98-3:2008, G.1.5], for carrying out the calculations required as part of an evaluation of measurement uncertainty. The approach applies to arbitrary models having a single output quantity where the input quantities are characterized by any specified PDFs [ISO/IEC Guide 98-3:2008, G.1.4, G.5.3].
As in the GUM, this Supplement is primarily concerned with the expression of uncertainty in the measurement of a well-defined physical quantity—the measurand—that can be characterized by an essentially unique value [ISO/IEC Guide 98-3:2008, 1.2].
This Supplement also provides guidance in situations where the conditions for the GUM uncertainty framework [ISO/IEC Guide 98-3:2008, G.6.6] are not fulfilled, or it is unclear whether they are fulfilled. It can be used when it is difficult to apply the GUM uncertainty framework, because of the complexity of the model, for example. Guidance is given in a form suitable for computer implementation.
This Supplement can be used to provide (a representation of) the PDF for the output quantity from which
a) an estimate of the output quantity,
b) the standard uncertainty associated with this estimate, and
c) a coverage interval for that quantity, corresponding to a specified coverage probability
can be obtained.
Given (i) the model relating the input quantities and the output quantity and (ii) the PDFs characterizing the input quantities, there is a unique PDF for the output quantity. Generally, the latter PDF cannot be determined analytically. Therefore, the objective of the approach described here is to determine a), b), and c) above to a prescribed numerical tolerance, without making unquantified approximations.
For a prescribed coverage probability, this Supplement can be used to provide any required coverage interval, including the probabilistically symmetric coverage interval and the shortest coverage interval.
This Supplement applies to input quantities that are independent, where each such quantity is assigned an appropriate PDF, or not independent, i.e. when some or all of these quantities are assigned a joint PDF.
Typical of the uncertainty evaluation problems to which this Supplement can be applied include those in which ⎯ the contributory uncertainties are not of approximately the same magnitude [ISO/IEC Guide 98-3:2008, G.2.2],
⎯ it is difficult or inconvenient to provide the partial derivatives of the model, as needed by the law of propagation of uncertainty [ISO/IEC Guide 98-3:2008, Clause 5],
⎯ the PDF for the output quantity is not a Gaussian distribution or a scaled and shifted t-distribution [ISO/IEC Guide 98-3:2008, G.6.5],
⎯ an estimate of the output quantity and the associated standard uncertainty are approximately of the same magnitude [ISO/IEC Guide 98-3:2008, G.2.1],
⎯ the models are arbitrarily complicated [ISO/IEC Guide 98-3:2008, G.1.5], and
⎯ the PDFs for the input quantities are asymmetric [ISO/IEC Guide 98-3:2008, G.5.3].
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A validation procedure is provided to check whether the GUM uncertainty framework is applicable. The GUM uncertainty framework remains the primary approach to uncertainty evaluation in circumstances where it is demonstrably applicable.
It is usually sufficient to report measurement uncertainty to one or perhaps two significant decimal digits.
Guidance is provided on carrying out the calculation to give reasonable assurance that in terms of the information provided the reported decimal digits are correct.
Detailed examples illustrate the guidance provided.
This document is a Supplement to the GUM and is to be used in conjunction with it. Other approaches generally consistent with the GUM may alternatively be used. The audience of this Supplement is that of the GUM.
NOTE 1 This Supplement does not consider models that do not define the output quantity uniquely (for example, involving the solution of a quadratic equation, without specifying which root is to be taken).
NOTE 2 This Supplement does not consider the case where a prior PDF for the output quantity is available, but the treatment here can be adapted to cover this case [16].
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.
ISO/IEC Guide 98-3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) ISO/IEC Guide 99:2007, International vocabulary of metrology — Basic and general concepts and associated terms (VIM)
3 Terms and definitions
For the purposes of this document, the terms and definitions of the ISO/IEC Guide 98-3 and the ISO/IEC Guide 99 apply unless otherwise indicated. Some of the most relevant definitions, adapted where necessary from these documents (see 4.2), are given below. Further definitions are given, including definitions taken or adapted from other sources, that are important for this Supplement.
A glossary of principal symbols is given in Annex G.
3.1
probability distribution
〈random variable〉 function giving the probability that a random variable takes any given value or belongs to a given set of values
NOTE The probability on the whole set of values of the random variable equals 1.
[Adapted from ISO 3534-1:1993, 1.3; ISO/IEC Guide 98-3:2008, C.2.3]
3.12
coverage interval
interval containing the value of a quantity with a stated probability, based on the information available
NOTE 1 A coverage interval is sometimes known as a credible interval or a Bayesian interval.
NOTE 2 Generally there is more than one coverage interval for a stated probability.
NOTE 3 A coverage interval should not be termed ‘confidence interval’ to avoid confusion with the statistical concept [ISO/IEC Guide 98-3:2008, 6.2.2].
NOTE 4 This definition differs from that in the ISO/IEC Guide 99:2007, since the term ‘true value’ has not been used in this Supplement, for reasons given in the GUM [ISO/IEC Guide 98-3:2008, E.5].
3.13
coverage probability
probability that the value of a quantity is contained within a specified coverage interval
NOTE The coverage probability is sometimes termed “level of confidence” [ISO/IEC Guide 98-3:2008, 6.2.2].
3.14
length of a coverage interval
largest value minus smallest value in a coverage interval
3.15
probabilistically symmetric coverage interval
coverage interval for a quantity such that the probability that the quantity is less than the smallest value in the interval is equal to the probability that the quantity is greater than the largest value in the interval
3.16
shortest coverage interval
coverage interval for a quantity with the shortest length among all coverage intervals for that quantity having the same coverage probability
3.17
propagation of distributions
method used to determine the probability distribution for an output quantity from the probability distributions assigned to the input quantities on which the output quantity depends
NOTE The method may be analytical or numerical, exact or approximate.
3.18
GUM uncertainty framework
application of the law of propagation of uncertainty and the characterization of the output quantity by a Gaussian distribution or a scaled and shifted t-distribution in order to provide a coverage interval
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4.7 The PDF for Y is denoted by gY (η) and the distribution function for Y by GY (η).
4.8 In the body of this Supplement, a quantity is generally denoted by an upper case letter and the expectation of the quantity or an estimate of the quantity by the corresponding lower case letter. For example, the expectation or an estimate of a quantity Y would be denoted by y. Such a notation is largely inappropriate for physical quantities, because of the established use of specific symbols, e.g. T for temperature and t for time. Therefore, in some of the examples (Clause 9), a different notation is used. There, a quantity is denoted
by its conventional symbol and its expectation or an estimate of it by that symbol hatted. For instance, the quantity representing the deviation of the length of a gauge block being calibrated from nominal length (9.5) is denoted by δL and an estimate of δL by δL
NOTE A hatted symbol is generally used in the statistical literature to denote an estimate.
4.9 In this Supplement, the term “law of propagation of uncertainty” applies to the use of a first-order Taylor series approximation to the model. The term is qualified accordingly when a higher-order approximation is used.
4.10 The subscript “c” [ISO/IEC Guide 98-3:2008, 5.1.1] for the combined standard uncertainty is redundant in this Supplement. The standard uncertainty associated with an estimate y of an output quantity Y can therefore be written as u(y), but the use of uc(y) remains acceptable if it is helpful to emphasize the fact that it represents a combined standard uncertainty. The qualifier “combined” in this context is also regarded as superfluous and may be omitted: the presence of “y” in “u(y)” already indicates the estimate with which the standard uncertainty is associated. Moreover, when the results of one or more uncertainty evaluations become inputs to a subsequent uncertainty evaluation, the use of the subscript “c” and the qualifier “combined” are then inappropriate.
4.11 The terms “coverage interval” and “coverage probability” are used throughout this Supplement. The GUM uses the term “level of confidence” as a synonym for coverage probability, drawing a distinction between “level of confidence” and “confidence level” [ISO/IEC Guide 98-3:2008, 6.2.2], because the latter has a specific definition in statistics. Since, in some languages, the translation from English of these two terms yields
the same expression, the use of these terms is avoided here.
4.12 According to Resolution 10 of the 22nd CGPM (2003) “ … the symbol for the decimal marker shall be either the point on the line or the comma on the line …”.
Exceptionally, for the decimal sign in this Guide 98 series, it has been decided to adopt the point on the line in the English texts and the comma on the line in the French texts.
4.13 Unless otherwise qualified, numbers are expressed in a manner that indicates the number of meaningful significant decimal digits.
EXAMPLE The numbers 0.060, 0.60, 6.0 and 60 are expressed to two significant decimal digits. The numbers 0.06, 0.6, 6 and 6 × 101 are expressed to one significant decimal digit. It would be incorrect to express 6 × 101 as 60, since two significant decimal digits would be implied.
4.14 Some symbols have more than one meaning in this Supplement. See Annex G. The context clarifies the usage.
4.15 The following abbreviations are used in this Supplement:
CGPM Conférence Générale des Poids et Mesures
IEEE Institute of Electrical and Electronic Engineers
GUF GUM uncertainty framework
JCGM Joint Committee for Guides in Metrology
GUM Guide to the expression of uncertainty in measurement
MCM Monte Carlo method
PDF probability density function
VIM International vocabulary of basic and general terms in metrology
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5 Basic principles
5.1 Main stages of uncertainty evaluation
5.1.1 The main stages of uncertainty evaluation constitute formulation, propagation, and summarizing:
a) Formulation:
1) define the output quantity Y, the quantity intended to be measured (the measurand);
2) determine the input quantities X = (X1,…,XN)T upon which Y depends;
3) develop a model relating Y and X;
4) on the basis of available knowledge, assign PDFs—Gaussian (normal), rectangular (uniform), etc.— to the Xi. Assign instead a joint PDF to those Xi that are not independent;
b) Propagation:
propagate the PDFs for the Xi through the model to obtain the PDF for Y;
c) Summarizing:
use the PDF for Y to obtain
1) the expectation of Y, taken as an estimate y of the quantity,
2) the standard deviation of Y, taken as the standard uncertainty u(y) associated with y
[ISO/IEC Guide 98-3:2008, E.3.2], and
3) a coverage interval containing Y with a specified probability (the coverage probability).
NOTE 1 The expectation may not be appropriate for all applications (cf. [ISO/IEC Guide 98-3:2008, 4.1.4]).
NOTE 2 The quantities described by some distributions, such as the Cauchy distribution, have no expectation or standard deviation. A coverage interval for the output quantity can always be obtained, however.
5.1.2 The GUM uncertainty framework does not explicitly refer to the assignment of PDFs to the input quantities. However [ISO/IEC Guide 98-3:2008, 3.3.5], “… a Type A standard uncertainty is obtained from a probability density function … derived from an observed frequency distribution …, while a Type B standard uncertainty is obtained from an assumed probability density function based on the degree of belief that an event will occur …. Both approaches employ recognized interpretations of probability.”
NOTE The use of probability distributions in a Type B evaluation of uncertainty is a feature of Bayesian inference [21, 27]. Research continues [22] on the boundaries of validity for the assignment of degrees of freedom to a standard uncertainty based on the Welch-Satterthwaite formula.
5.1.3 The steps in the formulation stage are carried out by the metrologist, perhaps with expert support.
Guidance on the assignment of PDFs (step 4) of stage a) in 5.1.1) is given in this Supplement for some
common cases (6.4). The propagation and summarizing stages, b) and c) in 5.1.1, for which detailed guidance is provided here, require no further metrological information, and in principle can be carried out to any required numerical tolerance for the problem specified in the formulation stage.
NOTE Once the formulation stage a) in 5.1.1 has been carried out, the PDF for the output quantity is completely specified mathematically, but generally the calculation of the expectation, standard deviation and coverage intervals requires numerical methods that involve a degree of approximation.
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